where, \(\gamma \)  is the ratio of specific heat of a gas. The conservation laws in eq. (REF ) can be obtained by taking \(\Psi \)  - moments of the \(2D\)  Boltzmann equation in the Euler limit [1]}. In the inner product form, these equations can be related as
\(\frac{\partial U}{\partial t} + \frac{\partial Gx}{\partial x} + \frac{\partial Gy}{\partial y} = \left< \Psi , \frac{\partial F}{\partial t} + v_1 \frac{\partial F}{\partial x} + v_2 \frac{\partial F}{\partial y} \right> = 0\) 
